Effective Reynolds Number for a swept wing

[Murmur79]

Suppose we have an infinite straight wing, using a given airfoil. Also, suppose for simplicity the B.L. is completely turbulent, and M<<1 (incompressible fluid).

As we know, the forces per unit length are: L=q⋅c⋅c_l, D=q⋅c⋅c_d, where c_l and c_d are the coefficients of the 2D airfoil for the given Re and α.

Now, if we rotate the infinite wing of an angle Λ, we have an infinite swept wing.

The theory says that in this case, the forces per unit length (parallel to leading edge) become: L=q⋅cos²Λ⋅c⋅c_l, D=q⋅cos²Λ⋅c⋅c_d.

Here is my question:

when looking up the c_l and c_d for the 2D airfoil, should we use:

.) the Re for the unswept wing: Re=U⋅c/ν

.) the Re normal to leading edge: Re=U⋅cosΛ⋅c/ν

.) the Re parallel to the flow: Re=U⋅(c/cosΛ)/ν

 

[Murmur79]

Ok, meanwhile a simpler example came to my mind.

Consider a flat plate of infinite length and chord c at zero incidence. Incompressible flow and 100% turbulent B.L.

In this case, we know that, for example, the thickness of the B.L. at the trailing edge will be δ=f(Re).

Now we rotate the flat plate of angle Λ.

What will the new B.L. thickness be at the trailing edge? What Re does it make sense to use between the 3 options given above?

 

[boneh3ad]

I'll make a few comments here. First, a swept flat plat behaves very differently than a swept wing. The latter has spanwise pressure gradients. The former does not.

Second, for a wing swept at a fixed $\Lambda$, do you expect the Reynolds number trends to change whether it (a constant) is included in the Reynolds number or not?

 

[Murmur79]

hi bone3ead,

yes, I realized the two cases are different. Let's consider the simpler example of the flat plate. In this case, the spanwise/chordwise gradients should be minimized.

I was wondering, for the experiment described above, which one between the three Re definitions, would give the closest results to the actual B.L. thickness at the trailing edge in the swept flat plate, _in the hypothesis_ that we use the same formula for the B.L. thickness in the two cases (swept/unswept).

I know that using the same formula has no theoretical basis at all, and could (intuitively) only be reasonable if the B.L. characteristics are not much changed, hence the hypothesis of the flat plate which should minimize spanwise effects.

In other words, I'd be curious to know what would happen at the thickness of the B.L. if we sweep a flat plate from 0 to say 45 or 60 degrees, and if there is a new specific Re that can predict the result without changing the formula used for the unswept flat plate.